how to make UIList interactive for object manipulation
In draw_item() you pass item and then use if item.name in bpy.data.objects.keys() - if you rename a key in bpy.data.objects and it no longer compares to item.name then that tells me that item is at least a copy or part copy of the data in bpy.data.objects.

This means the fix will be to update the name in both bpy.data.objects and item.

The fix will depend on what is in the rest of your code. Maybe an extra refresh of the data in item. One possibility is using bpy.props, by utilising the update feature of bpy.props you can have a small function that will update both your item data and the objects name property, that would mean having the list show your list of item properties not bpy.data.object properties.